Boundaries of quantum supremacy via random circuit sampling

Google's recent quantum supremacy experiment heralded a transition point where quantum computing performed a computational task, random circuit sampling, that is beyond the practical reach of modern supercomputers. We examine the constraints of the observed quantum runtime advantage in an extrapolation to circuits with a larger number of qubits and gates. Due to the exponential decrease of the experimental fidelity with the number of qubits and gates, we demonstrate for current fidelities a theoretical classical runtime advantage for circuits deeper than a few hundred gates, while quantum runtimes for cross-entropy benchmarking limit the region of a quantum advantage to a few hundred qubits. However, the quantum runtime advantage boundary in circuit width and depth grows exponentially with respect to reduced error rates, and our work highlights the importance of continued progress along this line. Extrapolations of measured error rates suggest that the limiting circuit size for which a computationally feasible quantum runtime advantage in cross-entropy benchmarking can be achieved approximately coincides with expectations for early implementations of the surface code and other quantum error correction methods. Thus the boundaries of quantum supremacy via random circuit sampling may fortuitously coincide with the advent of scalable, error corrected quantum computing in the near term.